22. How many positive integers less than 1000 a) are divisible by 7? b) are divisible by 7 but not by 11? c) are divisible by both 7 and 11? d) are divisible by either 7 or 11? e) are divisible by exactly one of 7 and 11? f ) are divisible by neither 7 nor 11? g) have distinct digits? h) have distinct digits and are even?

The m<3 is approximately equal to 85.39 degrees.

The sum of the measures of the angles of a triangle is 180 degrees. So, we have:

m<1 + m<2 + m<3 = 180

So, (3x + 12) + (3x + 18) + (7x + 10) = 180

Simplifying and solving for x, we get:

13x + 40 = 180

13x = 140

x = 10.77

Now, we can find m<3 by substituting x in the expression for m<3:

m<3 = 7x + 10

m<3 = 7(10.77) + 10

m<3 = 85.39

Therefore, m<3 is approximately equal to 85.39 degrees.

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The solution of the differential equation y'' + 2y' - y = xe is found to be [tex]y = Ce^{(-1-\frac{\sqrt{6}}{2})t} + C'e^{(-1+\frac{\sqrt{6}}{2})t} - \frac{4}{3}xe^{x} + Ce^{x}[/tex]

To solve the differential equation y'' + 2y' - y = xeˣ, we first find the complementary solution by solving the characteristic equation,

r² + 2r - 1 = 0

Using the quadratic formula, we get,

r = (-2 ± √(2² + 4(1)(1))) / 2

r = (-2 ± √6) / 2

So the roots are,

r₁ = -1 - √6/2

r₂ = -1 + √6/2

Therefore, the complementary solution is:

[tex]y_c = Ce^{r_{1}t} + C'e^{r_{2}t}\\y = Ce^{(-1-\frac{\sqrt{6}}{2})t} + C'e^{(-1+\frac{\sqrt{6}}{2})t}[/tex]

To find the particular solution, we use the method of undetermined coefficients. We assume a particular solution of the form,

y_p = Ax²eˣ + Bxeˣ + Ceˣ

Taking the first and second derivatives, we get,

y'_p = (2Ax + B + A)x eˣ + Beˣ + Ceˣ

y''_p = (4A + 2Ax + B + A)x eˣ + (2A + B)eˣ + Ceˣ

Substituting these expressions into the differential equation, we get,

(4A + 2Ax + B + A)x eˣ + (2A + B)eˣ + Ceˣ + 2[(2Ax + B + A)x eˣ + Beˣ + Ceˣ] - (Ax²eˣ + Bxeˣ + Ceˣ) = xeˣ

Simplifying, we get,

(3Ax² + 4Ax + B)x eˣ = xeˣ

Equating coefficients, we get,

3A = 1

4A + B = 0

Solving for A and B, we get,

A = 1/3

B = -4/3

Substituting these values into the assumed particular solution, we get,

y_p = (1/3)x²eˣ - (4/3)xeˣ + Ceˣ

Taking the first and second derivatives, we get,

y'_p = (2/3)x eˣ - 4/3eˣ + Ceˣ

y''_p = (2/3)eˣ + Ceˣ

Substituting the particular solution and its derivatives into the differential equation and simplifying, We get,

(2/3)xeˣ = xeˣ

This equation is true for all x, so there is no restriction on C. Therefore, the general solution is,

y = y_c + y_p

[tex]y = Ce^{(-1-\frac{\sqrt{6}}{2})t} + C'e^{(-1+\frac{\sqrt{6}}{2})t} - \frac{4}{3}xe^{x} + Ce^{x}[/tex]

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Complete question - y'' + 2y' - y = xeˣ solve the differential equation.

30 gallons of 28% butterfat and 30 gallons of 16% butterfat should be combined to create a 60-gallon mixture that is 22 % butterfat.

Let x gallon of 28% butterfat mixed with the y gallon of 16% butterfat to obtain 60 gallons of 22.5% butterfat,

⇒ x + y = 60 ⇒ x = 60 - y ----(1),

Quantity of butterfat in x gallon + quantity of butterfat in y gallon = total quantity of butterfat,

⇒ 28% of x + 16% of y = 22% of 60

⇒ 0.28x + 0.16y = 0.22 × 60

⇒ 0.28x + 0.16y = 13.2

⇒ 28x + 16y = 1320

⇒ 14x + 8y = 660

From equation (1),

14(60-y) + 8y = 660

840 - 14y + 8y = 660

6y = 180

y = 30

Again from equation (1),

x = 60 - y

= 60 - 30

= 30

Hence, 30 gallons of 28% butterfat and 30 gallons of 16% butterfat are combined to create a 60-gallon mixture that is 22%

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The constraint of x for the function g(x) = -40(2x + 7)(x - 10) is given by, 0 [tex]\le[/tex] x [tex]\le[/tex]6.

Given that the function is,

g(x) = - 40(2x + 7)(x - 10)

where x stands for the number of years since the year 2003

and g(x) stands for the amount of sales of golf equipment in million dollar unit.

The mathematician used the data for the year span of 2003 - 2009.

Min of x = Number of years since 2003 when the year is 2003 = 2003 - 2003 = 0

Max of x = Number of years since 2003 when the year is 2009 = 2009 - 2003 = 6

So the constraint of x is, 0 [tex]\le[/tex] x [tex]\le[/tex] 6.

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Using the law of sines, the length of b, to the nearest tenth, is: 21.4.

The Law of Sines is a mathematical equation that relates the lengths of the sides of a triangle to the sines of its angles. It states that for any triangle with sides of length a, b, and c and angles A, B, and C (where C is the largest angle), the following relationship holds:

[tex]\sf \dfrac{a}{sin(A)}= \dfrac{b}{sin(B)} =\dfrac{c}{sin(C)}[/tex]

This equation allows you to solve for the unknown sides or angles of a triangle if you know the lengths of its sides and the measure of at least one of its angles. It is often used in geometry and trigonometry to find the unknown parts of a triangle when you have limited information about it.

Given that triangle ABC has the following measures:

[tex]\sf m\angle B = 32^\circ[/tex]

[tex]\sf m\angle C = 38^\circ[/tex]

[tex]\sf a = 22[/tex]

[tex]\sf m\angle A = 180 - 32 - 38 = 110^\circ \ \ [triangle \ sum \ theorem][/tex]

Applying the law of sines, we have:

[tex]\sf \dfrac{38}{sin \ 110} =\dfrac{b}{sin \ 32}[/tex]

Cross multiply:

[tex]\sf (sin \ 110)(b) = (sin \ 32)(38)[/tex]

[tex]\sf b = \dfrac{(sin \ 32 \times38)}{(sin \ 110)}[/tex]

[tex]\sf b = 21.4 \ \ (nearest \ tenth)[/tex].

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We can expect a student to guess 17 correct answers on a test with 68 four-choice multiple choice questions with probability 0.25.

Since the probability of guessing the correct answer to a four-choice multiple choice question is 0.25, the probability of guessing an incorrect answer is 1 - 0.25 = 0.75.

We can model the number of correct guesses as a binomial distribution with n = 68 (the number of trials) and p = 0.25 (the probability of success, i.e., guessing correctly).

The expected value of a binomial distribution is given by the formula:

E(X) = n * p

So, in this case, the expected number of correct answers that a student should guess on the test is:

E(X) = 68 * 0.25 = 17

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Mean - this is the average. Add up the numbers and divide by the count of the numbers in the data set.

11+13+14+16+17+18+20+20+21+24+25+29 = 228

There are 12 numbers, so divide by 12.

228/12 = 19. The mean is 19.

Median - - - this is the MIDDLE number. Your data is already in numerical order (hooray!) so there are 12 numbers. Since it's even, we'll take the average of the 2 middle numbers. The 6th number is 18 and the 7th number is 20. The average of these is 19. So the median is 19.

(Side note: yes, median and mean can be the same.)

Mode - - - this is repeat/most common numbers! 20 repeats. 20 is the mode.

Range - - - biggest number is 29, littlest is 11. The range is 29-11 = 18.

Answer:

Part a)

[tex]MG = \begin{pmatrix}1&k_1+9&4k_2+36\\ 2t_1+4&t_1k_1-17&-19t_1+3k_2-11\\ t_2+1&k_1-4t_2+5&t_2k_2-8\end{pmatrix}[/tex]

Part b)

[tex]t_1 = -2\\\\t_2 = -1\\\\k_1 = -9\\\\k_2 = -9\\[/tex]

Step-by-step explanation:

a) Computing MG

We have
[tex]M = \begin{pmatrix}1&4&5\\ \:t_1&3&-1\\ \:1&t_2&1\end{pmatrix}[/tex]

[tex]G = \begin{pmatrix}2&k_1&-19\\ \:1&-4&k_2\\ \:-1&5&11\end{pmatrix}[/tex]

[tex]MG = \begin{pmatrix}1&4&5\\ \:t_1&3&-1\\ \:1&t_2&1\end{pmatrix}\times \begin{pmatrix}2&k_1&-19\\ \:1&-4&k_2\\ \:-1&5&11\end{pmatrix}[/tex]

[tex]= \begin{pmatrix}1&k_1+9&4k_2+36\\ 2t_1+4&t_1k_1-17&-19t_1+3k_2-11\\ t_2+1&k_1-4t_2+5&t_2k_2-8\end{pmatrix}[/tex]

I am using a matrix calculator to compute the product rather than going through the rather cumbersome multiplication exercise by hand

b) One property of matrices is that if M is a matrix and M⁻¹ is its inverse
M x M⁻¹ is the identity matrix which has a 1 along the primary diagonal (top left - bottom right) and zeros everywhere

Here we are given G = M⁻¹ so M x G is an identity matrix

[tex]MG(1, 2) \\= k_1 + 9 = 0\\= > k_1 = -9\\\\MG(1, 3)\; is\\ 4k_2 + 36 = 0\\k_2 = -36/4 = -9\\\\MG(2, 1) = 2t_1 + 4 = 0\\\implies t_1 = -4/2 = -2\\\\MG(3, 1) = t_2 + 1\\t_2 + 1 = 0\\t_2 = -1\\[/tex]

So the values of the variables are

[tex]t_1 = -2\\\\t_2 = -1\\\\k_1 = -9\\\\k_2 = -9\\[/tex]

To check if these are correct, you can plug in these values into the product matrix MG. You will see that all elements except the diagonal elements are 0; diagonal elements are 1

Step-by-step explanation:

In the side view diagram, we can see that the tank, the wall, and the floor form a right triangle with the radius of the tank as the hypotenuse. Let x be the distance between the tank and the wall, as shown below:

```

|\

| \

r | \ x

| \

|_ _\

s

```

Using the Pythagorean theorem, we can write:

r^2 = x^2 + s^2

Rearranging this equation, we get:

x^2 = r^2 - s^2

Subtracting r^2 from both sides, we get:

x^2 - r^2 = -s^2

Multiplying both sides by -1, we get:

s^2 - x^2 = 0

Therefore, the equation for the relationship between the two radii is:

s^2 - x^2 = 0

Simplifying this equation, we get:

s^2 = x^2

or

x^2 = s^2

So, there is a zero on one side of the equals sign.

The quadratic function f(x) = 2x² - 5x + 3, it is found :

a. The factored function is, f(x) = 2(x - 1)(x - 1.5).

b. The x-intercepts of the function are: x = 1, x = 1.5.

c. The end behavior of the graph is:

As x -> -∞, f(x) -> -∞.

As x -> ∞, f(x) -> ∞.

The quadratic function is :

f(x) = 2x² - 5x + 3.

Hence the coefficients are:

a = 2, b = -5, c = 3.

Using a calculator with these coefficients, the x-intercepts are:

x = 1.

x = 1.5.

Then the factored form of the function;

f(x) = 2(x - 1)(x - 1.5).

Let the leading coefficient of 2 and the x-intercepts.

This is a concave-up parabola, it goes to positive infinity to the left and to the right of the graph.

Hence the end behavior is;

As x -> -∞, f(x) -> -∞.

As x -> ∞, f(x) -> ∞.

To determine the graph, the axis of symmetry is:

x = -b/2a = 5/(2(2)) = 1.25.

The y-coordinate at this point, that is the turning point of the function, is:

f(1.25) = 2(1.25)² - 5(1.25) + 3 = -0.125.

The y-intercept of the function

f(0) = 2(0)² - 5(0) + 3 = 3.

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Answer:

50(5π + 6) cm²

Step-by-step explanation:

TSA of figure = 1/2 of TSA(Cylinder) + Area of rectangle

= 1/2 of 2πr(h+r) + 20*15

= πr(h+r) + 300

= 10π(15+10) + 300

= 250π + 300 cm²

or 50(5π + 6) cm²

1. The product of the matrix MG would be [tex]\left[\begin{array}{ccc}1&k_1+9&4k_2+36\\2t_1&k_1t_1-17&3k_2-19t_1-11\\t_2+1&k_1-4t_2+5&k_2t_2-8\end{array}\right][/tex]

2. The values of k₁ = -9;  k₂ = -9; t₁ = 0; t₂ = -1

The values of the matrix M and G are;

M = [1, 4, 5;                       G = [2, k₁, -19;

      t₁, 3, -1;                             1, -4, k₂;

      1, t₂, 1]                              -1, 5, 11]

The product MG will be row 1

1×2 + 4×1 + 5×-1     1k₁+4×-4+ 5×5       1×-19+4k₂+5×11 ⇒1    k₁+9    4k₂+36

Row 2

t₁×2+3×1+-1×-1    t₁k₁+3×-4+-1×5       t₁×-19+3k₂+-1×11  ⇒ 2t₁    k₁t₁-17 3k₂-19t₁-11

Row 3

1×2+t₂×1+1×-1      1k₁+t₂×-4+-1×5      1×-19+t₂k₂+1×11 ⇒ t₂+1   k₁-4t₂+5    k₂t₂-8

Therefore MG =

1     k₁+9   4k₂+36

2t₁    k₁t₁-17    3k₂-19t₁-11

t₂+1   k₁-4t₂+5      k₂t₂-8

G = M⁻¹  should be an identity matrix which looks like

IM = [ 1 0 0

         0 1 0

        0 0 1 ]

MG and I to find the values of t₁, t₂, k₁, and k₂:

1 = 1,         k₁+9 = 0,       4k₂+36 = 0

2t₁ = 0     k₁t₁-17 = 1        3k₂-19t₁-11 = 0

t₂+1 = 0     k₁-4t₂+5 = 0   k₂t₂-8 = 1

k₁+9 = 0 ⇒k₁ = -9          

4k₂+36 = 0  ⇒  k₂ = -9          

2t₁ = 0 ⇒   t₁ = 0          

t₂+1 = 0 ⇒    t₂ = -1

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Answer:

2m=7. m=7/2. 10x =48. x = 4.8

Step-by-step explanation:

2m

Answer:

a)  7/2 or 3.5

b) 4.8

form equation and solve them

a}n = 17

Step-by-step explanation:

a) 2m + 8 = 15

Subtract 8 from both sides: 2m + 8 - 8 = 15 - 8

Simplify: 2m = 7

Divide both sides by 2: (2m)/2 = 7/2

Simplify: m = 7/2 or 3.5

b) 10x - 3 = 45

Add 3 to both sides: 10x - 3 + 3 = 45 + 3

Simplify: 10x = 48

Divide both sides by 10: (10x)/10 = 48/10

Simplify: x = 4.8

To form and solve an equation for Ashley’s problem, we need to translate the words into mathematical expressions. Here are the steps and solution:

a) Ashley thinks of a number if i double my number and add 6 the answer is 40

Let n be Ashley’s number.

If we double the number, we get 2n.

If we add 6 to the doubled number, we get 2n + 6.

The answer is equal to 40, so we can write an equation: 2n + 6 = 40

To solve for n, we need to isolate it on one side of the equation by using inverse operations.

Subtract 6 from both sides: 2n + 6 - 6 = 40 - 6

Simplify: 2n = 34

Divide both sides by 2: (2n)/2 = 34/2

Simplify: n = 17

________________________________

________________________________

Just add the powers while it is multiply

-1+(-3)

= -1 - 3

= -4

So it is [tex]2^{-4}[/tex]

Answer:

C) 16

Step-by-step explanation:

(3x + 5) + 37 = 90

3x = 90 - 5 - 37

3x = 48

x = 48/3 = 16

Answer:

x = 16°

Step-by-step explanation:

It is 90° angle so their sum must be 90°( complement) .

3x + 5° + 37° = 90°

3x + 42° = 90° ... simplify the like term

3x = 90° - 42°

3x = 48°

3x/3 = 48°/3

x = 16°

So the answer is C, 16

The presence of the coefficient "a" in ax^2 + bx + c requires that we look for numbers that have a product of ac

Here we have to determine the difference

From the question, we have the following parameters that can be used in our computation:

ax^2 + bx +c and x^2 + bx +c

The presence of the coefficient "a" in the first expression makes a difference in the factoring process.

When factoring x^2 + bx + c, we are looking for two numbers that multiply to give c and add up to b.

For ax^2 + bx +c , we look for the product of ac and sum of b

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The solution set of the inequality (2/5)*(10x - 15) ≥ 10 is:

[4, ∞)

Here we want to solve the inequality:

(2/5)*(10x - 15) ≥ 10

To solve this we need to isolate the variable x, to do so, we can start multiplying both sides by 5/2

(10x - 15) ≥ 10*(5/2)

10x - 15 ≥ 25

Now add 15 in both sides:

10x ≥ 25 + 15

Now divide both sides by 10.

x ≥ 40/10

x ≥ 4

So the solution set is the set of all real numbers equal or larger than 4.

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The surface area of the box increases by 67.26 cm² when the height is increased by 0.7 cm.

Recall the formula, Surface Area (SA) of the rectangular prism is given as:

SA = 2lw + 2lh + 2wh

where

l = length,

w = width,

h =  height

From the diagram, we know that:

l = 18cm

w = 6.3cm

h = 30cm

Substituting these values into the surface area formula, we get:

SA₁ = 2(18)(6.3) + 2(18)(30) + 2(6.3)(30) = 2376 cm²

When the height is increased by 0.7cm, the new height becomes:

h' = h + 0.7cm = 30.7cm

The new surface area can be calculated using the same formula:

SA₂ = 2(18)(6.3) + 2(18)(30.7) + 2(6.3)(30.7) = 2443.26 cm²

The increase in surface area is the difference between the new surface area and the original surface area:

SA₂ - SA₁ = 2443.26 cm² - 2376 cm²

= 67.26 cm²

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Answer:

28.57% chance

Step-by-step explanation:

add them all up which is 14, and if there are 4 white marbles that means its 4/14, which is 28.57 as a percent (2:7 as ratio)

To memorize knowledge there are several tips that can help us. Some are: Repeat some knowledge from time to time to memorize them.

To memorize knowledge there are several methods that can be useful when we need to learn something difficult. In this case, we must use some methods that can help us such as:

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The two values for x so the points form a right triangle are x = -4 and x = 10

From the question, we have the following parameters that can be used in our computation:

The coordinates of the points are A(5,3), B(5,6) and C(x,3).

The third point is given as

C(x,3)

The value of x can be any real value apart from 5

This is because if x = 5, then it would be the same as point A

So, we can use the following values for x

x = -4 and x = 10

Note that all real values work for x

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Answer:

Right three

Step-by-step explanation:

Because the translation is applied inside the parentheses, we are shifting the function left/right. To determine whether it is shifted left or right, note that shifting right will have a negative number and shifting left will have a positive number. Since we have a negative number -3, it is shifted right.

Answer:

Step-by-step explanation:

Here if we put [tex]x=3[/tex] then then equation will give 0. Hence it will shift right side three unit .

Final answer: right 3

The value of transformation of f (x) makes f(x) < g(x) is,

⇒ shifts the function 3 units upward.

We know that;

A transformation that occurs when a figure is moved from one location to another location without changing its size or shape is called translation.

Now, We have;

Functions are,

f (x) = x²

g(x) = x² +3

Hence, The value of transformation of f (x) makes f(x) < g(x) is,

⇒ shifts the function 3 units upward.

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The distance between the fire and station B is 10.7miles

The sine rule states that if a, b and c are the lengths of the sides of a triangle, and A, B and C are the angles in the triangle; with A opposite a, etc., then a/sinA=b/sinB=c/sinC.

The angle at A = 90- 35

= 55°

The angle at B = 90-49

= 41°

Angle at the fire = 180-(41+55)

= 180-96 = 84°

Using sine rule

sin84/13 = sin55/x

xsin84 = 13sin55

0.995x = 10.65

x = 10.65/0.995

x = 10.7 miles

Therefore the distance between the fire and station B is 10.7 miles.

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Answer:

Step-by-step explanation:

a.) When we reflect over xz plane x and z coordinates stay same, y coordinate  changes to same numerical value but opposite sign. Moving front-back is moving over x-axis, moving left-right is moving over y-axis, moving up-down is moving over z-axis.

A(0,0,0)

Reflecting

A(0,0,0)

B(0,5,0)

Reflecting

B(0,-5,0)

C(3,5,0)

Reflecting

C(3,-5,0)

D(3,0,0)

Reflecting

D(3,0,0)

b.)

A(0,0,0)

Moving

A(-2,-3,1)

B(0,-5,0)

Moving

B(-2,-8,1)

C(3,-5,0)

Moving

C(1,-8,1)

D(3,0,0)

Moving

D(1,-3,1)

The values of the complex expressions are ✓(x - iy) = a - ib and x² + y² =  (a + ib)²(a - ib)²

From the question, we have the following parameters that can be used in our computation:

✓(x + iy) = a + ib

Changing the signs, we have

✓(x - iy) = a - ib

Multiply both expressions

This gives

✓(x + iy) * ✓(x - iy) =  (a + ib)(a - ib)

Square both sides

So, we have

(x + iy) * (x - iy) =  (a + ib)²(a - ib)²

This gives

x² + y² =  (a + ib)²(a - ib)²

Hence, the values of ✓(x - iy) is a - ib and x² + y² is  (a + ib)²(a - ib)²

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We can predict that there are approximately 2,250 Art majors in the total population of 7,500 students at ASU based on the sample data.

We have,

We can use the formula for estimating population proportion:

Population proportion = sample proportion x population size

where the sample proportion is the fraction of students in the sample who are Art majors, and the population size is the total number of students in the population.

From the sample of 250 students, 75 said that they are majoring in Art, so the sample proportion of Art majors.

sample proportion = 75/250 = 0.3

The population size is 7,500 students.

Using the formula above, we can estimate the number of Art majors in the total population:

population proportion = sample proportion x population size

population proportion = 0.3 x 7,500

population proportion = 2,250

Therefore,

We can predict that there are approximately 2,250 Art majors in the total population of 7,500 ASU students based on the sample of 250 students surveyed.

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The total number of ways to arrange all 6 tables is 720

There are 15 ways to choose 4 tables out of 6.

(a) There are 6 options for the first table, 5 options left for the second table, 4 options left for the third table, and so on.

Therefore, the total number of ways to arrange all 6 tables is:

6 x 5 x 4 x 3 x 2 x 1 = 720

(b) To choose 4 tables out of 6, we can use the combination formula:

C(6,4) = 6! / (4! x 2!)

= (6 x 5 x 4 x 3) / (4 x 3 x 2 x 1)

= 15

Therefore, there are 15 ways to choose 4 tables out of 6.

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